Circuit Depth Reduction of One-Ancilla Quantum Differential Equation Solver via Extrapolation

Jul 16, 2026

Solving linear differential equations is a core task in scientific computing and an important primitive for quantum computing. A one-ancilla quantum differential equation solver offers a hardware-friendly, locality-preserving approach with provable performance, but its maximum single-run circuit depth grows as O(1/ε) with target accuracy ε. This work reduces depth by combining the solver with classical step-size postprocessing, running it at logarithmically many time steps and canceling leading discretization errors. The result lowers maximum depth to O(polylog(1/ε)) without extra quantum ancillae or locality loss, using holomorphic extension techniques and verified via numerical experiments on the Hatano–Nelson model and convection–diffusion equation.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages28

Reading Time1 h 11 min

Introduction

Quantum differential equation solvers and resource trade-offs

One-ancilla dissipative algorithm and its circuit-depth dependence

Classical post-processing and extrapolation-based error mitigation

Goal and approach of this work

Main idea and technical ingredients

Extrapolation with logarithmically many time steps

Regularity estimates via holomorphic extension

Circuit Depth Reduction of One-Ancilla Quantum Differential Equation Solver via Extrapolation Di Fanga , Justin Parkb a Department of Mathematics and Duke Quantum Center, Duke University, Durham, NC, USA b Department of Mathematics and Department of Computer Science, Duke University, Durham, NC, USA 8 Jul 2026 | ARTICLE INFO | | AB STRACT | | --- | --- | --- | | Keywords:<br>quantum differential equations<br>quantum algorithms one-ancilla<br>qubit-efficient extrapolation classical postprocessing circuit depth | | Solving linear differential equations is a fundamental task in scientific computing and an important primitive for quantum computing. A recent one-ancilla quantum differential equationsolver provides a hardware-friendly and locality-preserving approach with provable performance guarantees, making it highly suitable for the early fault-tolerant and near-term regimes. Its simple circuit structure comes with a natural trade-off: the maximum single-run circuit depth scales as 􀁏(1∕􀀏) in the target accuracy 􀀏 . In this work, we reduce this depth by combining the solver with classical step-size postprocessing. By running the one-ancilla solver at a logarithmic number of finite time step sizes and using classical post-processing to cancel leading discretization errors, we reduce the maximum single-run circuit depth to 􀁏(polylog(1∕􀀏)) without adding quantum ancillae or sacrificing locality. Technically, extending extrapolation ideas beyond Hamiltonian and Lindbladian dynamics requires regularity estimates for observable maps under nonunitary evolution, which we obtain through a holomorphic extension ofthe adjoint evolution. Numerical experiments on the Hatano-Nelson model (ODE) and the convection-diffusion equation (PDE) demonstrate the effectiveness of the approach. | arXiv :2607 .07389v1 based on quantum linear systems algorithms with history-state encodings [1–6], time-marching methods [7, 8], linear combinations of Hamiltonian simulation [9–14], unitary-dilation [15–18], Lindbladian embeddings [19], quantum eigenvalue processing [20], and other fault-tolerant primitives. We remark that this dissipative regime is a standard setting shared across state-of-the-art quantum algorithmic frameworks for general linear differential equations, rather than an additional assumption specific to the present work or to a particular line of work. As in many areas of quantum algorithms, existing quantum differential equation solvers can be broadly viewed from two complementary perspectives. On the one hand, near-term algorithms are designed to be more compatible with current or early fault-tolerant hardware, often avoiding expensive coherent control, large ancilla registers, or deep circuit constructions. Variational and other NISQ approaches fall into this category, but rigorous performance guarantees are often difficult to establish in general. On the other hand, fully fault-tolerant quantum algorithms are developed with provable guarantees and can achieve excellent, sometimes optimal and near-optimal asymptotic scaling. However, these algorithms typically rely on advanced fully fault-tolerant primitives such as block-encodings, quantum linear systems algorithms (QLSA), quantum singular value transformation (QSVT), linear-combination-of-unitaries (LCU) techniques, compression gadget, or coherent control over many time steps. These ingredients can require many ancilla qubits and complicated circuit structures, making them less friendly to realize on near-term or early fault-tolerant devices. A recent algorithm in [21] provides a different point in this landscape. It gives a one-ancilla quantum algorithm for linear differential equations of the dissipative form considered in this work. The algorithm only uses local Hamiltonian simulation and mid-circuit measurements, preserves locality, and avoids advanced fault-tolerant subroutines such as LCU, QSVT, QLSA, or compression gadget. In this sense, it is both near-term frie

FAQs

What limits the circuit depth of the one-ancilla quantum differential equation solver?

Its maximum single-run circuit depth scales as O(1/ε), reflecting the first-order nature of the underlying quantum algorithm and the discretization accuracy requirement.

How does the proposed method reduce circuit depth without increasing quantum resources?

It runs the one-ancilla solver for a logarithmic number of finite time-step sizes and applies classical post-processing to cancel leading discretization errors, reducing depth to O(polylog(1/ε)) without adding quantum ancillae or sacrificing locality.

Why are holomorphic extension and adjoint evolution needed?

Extending extrapolation beyond Hamiltonian and Lindbladian dynamics requires regularity estimates for observable maps under nonunitary evolution, which are obtained through a holomorphic extension of the adjoint evolution.

Jul 16, 2026

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